I've been working as a contractor for a company for a few months now but I haven't signed a contract or a 1099 form, I just receive payment on services due when an invoice is submitted. If I start a small business would I be able to convert the earned money as money to the business instead of personally claiming it?
First off, the reason people call themselves contractors is because they sign a contract to do a job. Not doing so is risky for both you and your client. Get one today, that spells out the expectations, termination conditions, terms for payment, etc.
Next, you don't sign a 1099, you are given one at the end of the year. That is also sent to the IRS to report your untaxed income. If you haven't been paying the IRS estimated payments, then you need to start doing that, ASAP.
Now, for the small business, you didn't specify which type of small business. If you plan to become a sole proprietor, then you already are one. And there's no income to convert to your business since you are your business, one in the same. If you're talking about incorporating, then that's a whole different issue that involves equity, shares of ownership, etc.
@Lisa, the short answer is yes you can form a company and that company can earn the money instead of you directly. This would mean:
... Would you be able to convert your existing income? ... this is one to speak to an accountant about ... invoices have been done in your name so technically it wasn't your company that invoiced them ... that said if you setup the company before you do your FYE and speak to your clients about it you may be able to "retrospectively" take the income as your companies income instead.